Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-09T14:26:56.885Z Has data issue: false hasContentIssue false
  • English
  • Français

Dirichlet integral and Picard principle

Published online by Cambridge University Press:  22 January 2016

Mitsuru Nakai  and
Toshimasa Tada
Mitsuru Nakai
Affiliation:
Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa, Nagoya 466, Japan
Toshimasa Tada
Affiliation:
Department of Mathematics, Daido Institute of Technology, Doido, Minami, Nagoya 457, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

A density P on the punctured unit disk Ω:0 < |z| <1 is a 2-form P(z)dxdy whose coefficient P(z) is a real valued nonnegative locally Hölder continuous function on the closed punctured unit disk Ω:0< |z| <≦1. Here we consider Ω as an end of the punctured sphere 0 < |z| ≦ + + so that the point z = 0 is viewed as the ideal boundary δΣ of Σ and the unit circle |z| = 1 as the relative boundary δΣ of Σ. We denote by D = D(Σ) the family of densities on Σ.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 86 , June 1982 , pp. 85 - 99
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[ 1 ] Boukricha, A., Das Picard-Prinzip und verwandte Fragen bei Störung von harmonischen Räurnen, Math. Ann., 239 (1979), 247–270.CrossRefGoogle Scholar
[ 2 ] Brelot, M., Étude des l’équation de la chaleur Δu = c(M)u(M), c(M) ≧ 0, au voisinage d’un point singulier du coefficient, Ann. Sci. École Norm. Sup., 48 (1931), 153–246.CrossRefGoogle Scholar
[ 3 ] Brelot, M., Sur le principe des singularités positives et la notion de source pour l’équation (1) Δu(M) = c(M)u(M), (c ≧ 0), Ann. Univ. Lyon Sci. Math. Astro., 11 (1948), 9–19.Google Scholar
[ 4 ] Brelot, M., On Topologies and Boundaries in Potential Theory, Lecture Notes in Math., Springer, 1971.Google Scholar
[ 5 ] Constantinescu, C. und Cornea, A., Über einige Problem von M. Heins, Rev. Roumaine Math. Pures Appl., 4 (1959), 277–281.Google Scholar
[ 6 ] Constantinescu, C. und Cornea, A., Ideale Ränder Riemannscher Flächen, Springer, 1963.CrossRefGoogle Scholar
[ 7 ] Godefroid, M., Sur un article de Kawamura et Nakai à propos du principe Picard, Bull. Sci. Math., 102 (1978), 295–303.Google Scholar
[ 8 ] Hayashi, K., Les solutions positives de l’équation Δu =. Pu sur une surface de Riemann, Kōdai Math. Sem. Rep., 13 (1961), 20–24.Google Scholar
[ 9 ] Heins, M., Riemann surfaces of infinite genus, Ann. of Math., 55 (1952), 296–317.CrossRefGoogle Scholar
[10] Imai, H., On singular indices of rotation free densities, Pacific J. Math., 80 (1979), 179–190.CrossRefGoogle Scholar
[11] Imai, H. and Tada, T., Picard principle for rotation free densities on the Euclidean N-space (N ≧ 3), Bull. Daido Inst. Tech., 13 (1978), 1–12.Google Scholar
[12] Itô, S., Martin boundary for linear elliptic differential operator of second order in a manifold, J. Math. Soc. Japan, 16 (1964), 307–334.CrossRefGoogle Scholar
[13] Kawamura, M., Picard principle for finite densities on some end, Nagoya Math. J., 67 (1977), 35–40.CrossRefGoogle Scholar
[14] Kawamura, M., On a conjecture of Nakai on Picard principle, J. Math. Soc. Japan, 31 (1979), 359–372.CrossRefGoogle Scholar
[15] Kawamura, M., A remark on inhomogeneity of Picard principle, J. Math. Soc. Japan, 32 (1980), 517–519.CrossRefGoogle Scholar
[16] Kawamura, M. and Nakai, M., A test of Picard principle for rotation free densities II, J. Math. Soc. Japan, 28 (1976), 323–342.CrossRefGoogle Scholar
[17] Kuramochi, Z., An example of a null-boundary Riemann surface, Osaka Math. J., 6 (1954), 83–91.Google Scholar
[18] Lahtinen, A., On the existence of singular solutions of Δu = Pu on Riemann surfaces, Ann. Acad. Sci. Fenn., 546 (1973).Google Scholar
[19] Martin, R., Minimal positive harmonic functions, Trans. Amer. Math. Soc, 49 (1941), 137–172.CrossRefGoogle Scholar
[20] Nakai, M., The space of nonnegative solutions of the equation Δu = Pu on a Rieman surface, Kōdai Math. Sem. Rep., 12 (1960), 151–178.Google Scholar
[21] Nakai, M., Martin boundary over isolated singularity of rotation free density, J. Math. Soc. Japan, 26 (1974), 483–507.CrossRefGoogle Scholar
[22] Nakai, M., A test for Picard principle, Nagoya Math. J., 56 (1974), 105–119.CrossRefGoogle Scholar
[23] Nakai, M., A remark on Picard principle, Proc. Japan Acad., 50 (1974), 806–808.Google Scholar
[24] Nakai, M., A test of Picard principle for rotation free densities, J. Math. Soc. Japan., 27 (1975), 412–431.CrossRefGoogle Scholar
[25] Nakai, M., A remark on Picard principle II, Proc. Japan Acad., 51 (1975), 308–311.Google Scholar
[26] Nakai, M., Picard principle and Riemann theorem, Tôhoku Math. J., 28 (1976), 277–292.CrossRefGoogle Scholar
[27] Nakai, M., Picard principle for finite densities, Nagoya Math. J., 70 (1978), 7–14.CrossRefGoogle Scholar
[28] Nakai, M., Strong Picard principle, J. Math. Soc. Japan, 32 (1980), 631–638.CrossRefGoogle Scholar
[29] Ozawa, M., Some classes of positive solutions of Δu = Pu on Riemann surfaces,, I; II, Kōdai Math. Sem. Rep., 6 (1954); 7 (1955), 121–126; 15–20.Google Scholar
[30] Schiff, J. L., Nonnegative solutions of Δu = Pu on open Riemann surfaces, J. Analyse Math., 27 (1974), 230–241.CrossRefGoogle Scholar
[31] Sur, M., The Martin boundary for a linear elliptic second order operator, Izv. Akad. Nauk SSSR, 27 (1963), 45–60 (Russian).Google Scholar
[32] Tada, T., On a criterion of Picard principle for rotation free densities, J. Math. Soc. Japan, 32 (1980), 587–592.CrossRefGoogle Scholar